**How to Check if a Given Point Lies Inside or Outside a Circle :**

Here we are going to see, how to check if a given point lies inside or outside a circle.

x_{1}^{2} + y_{1}^{2} + 2gx_{1} + 2fy_{1} + c > 0 (outside of the circle)

x_{1}^{2} + y_{1}^{2} + 2gx_{1} + 2fy_{1} + c = 0 (on the circle)

x_{1}^{2} + y_{1}^{2} + 2gx_{1} + 2fy_{1} + c < 0 (inside the circle)

**Question 1 :**

Determine whether the points (-2, 1),(0, 0) and (-4, -3) lie outside, on or inside the circle x^{2} + y^{2} − 5x + 2y − 5 = 0 .

**Solution :**

Equation of the circle :

x^{2} + y^{2} − 5x + 2y − 5

Equation of the circle at the point (x_{1}, y_{1}) :

x_{1}^{2} + y_{1}^{2} − 5x_{1} + 2y_{1} − 5

at (-2, 1)

= (-2)^{2} + 1^{2} − 5(-2) + 2(1) − 5

= 5 + 10 + 2 - 5

= 12 > 0

So, the point (-2, 1) lies outside the circle.

at (0, 0)

= 0^{2} + 0^{2} − 5(0) + 2(0) − 5

= - 5 < 0

So, the point (0, 0) lies inside the circle.

at (-4, -3)

= (-4)^{2} + (-3)^{2} − 5(-4) + 2(-3) − 5

= 16 + 9 + 20 - 6 - 5

= 34 > 0

So, the point (-4, -3) lies outside the circle.

**Question 2 :**

Find centre and radius of the following circles.

(i) x^{2} + (y + 2)^{2} = 0

**Solution :**

By comparing the given equation with general form of a circle, we get

x^{2} + (y + 2)^{2} = 0

(x - h)^{2} + (y - k)^{2} = r^{2}

(h, k) ==> (0, -2) and r is 0.

Hence the center and radius of the circle are (0, -2) and 0.

(ii) x^{2} + y^{2} + 6x − 4y + 4 = 0

**Solution :**

Center of the circle is (-g, -f)

radius of the circle = √g^{2} + f^{2} - c

g = 3, f = -2 and c = 4

= √3^{2} + (-2)^{2} - 4

= √(9 + 4 - 4)

= √9

= 3

Hence the center and radius of the circle are (-3, 2) and 3 respectively.

(iii) x^{2} + y^{2} − x + 2y − 3 = 0

**Solution :**

g = -1/2, f = 1 and c = -3

= √(-1/2)^{2} + 1^{2} + 3

= √((1/4) + 4)

= √17/4

= √17/2

Hence the center and radius of the circle are (1/2, -1) and √17/2 respectively.

(iv) 2x^{2} + 2y^{2} − 6x + 4y + 2 = 0

**Solution :**

By dividing the entire equation by 2, we get

x^{2} + y^{2} − 3x + 2y + 1 = 0

g = -3/2, f = 1 and c = 1

= √(-3/2)^{2} + 1^{2} - 1

= √(9/4)

= 3/2

Hence the center and radius of the circle are (3/2, -1) and 3/2 respectively.

**Question 3 :**

If the equation 3x^{2} + (3 − p) xy + qy^{2} − 2 px = 8pq represents a circle, find p and q . Also determine the centre and radius of the circle.

**Solution :**

The equation x^{2} + y^{2} + 2gx + 2fy + c = 0 is a second degree equation in x and y possessing the following characteristics:

(i) It is a second degree equation in x and y ,

(ii) coefficient of x^{2} = coefficient of y^{2 }≠ 0,

(iii) coefficient of xy = 0 .

3x^{2} + (3 − p) xy + qy^{2} − 2 px = 8pq

According to (iii), coefficient of x y = 0

3 - p = 0

p = 3

According to (ii), coefficient of x^{2} = coefficient of y^{2 }≠ 0

coefficient of x^{2} = 3 = coefficient of y^{2 }

q = 3

By applying the values of p and q in the given equation, we get

3x^{2} + 3y^{2} − 6x = 72

x^{2} + y^{2} − 2x = 24

Center of the circle (-g, -f)

g = -1, f = 0 and c = -24

Center of the circle = (1, 0)

Radius of the circle = √g^{2} + f^{2} - c

= √(-1)^{2} + 0^{2} + 24

= √25 = 5

Hence the center and radius of the circle are (1, 0) and 5.

After having gone through the stuff given above, we hope that the students would have understood, "How to Check if a Given Point Lies Inside or Outside a Circle".

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

Widget is loading comments...

You can also visit our following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**